I think you'd have to narrow down "19th century" a bit, Cul - at the beginning of it, you could
buy a commission in a good regiment, and it was a favourite place for a younger son of the aristocracy - but at the other end of the century, the army was much more of of a meritocracy, with the purchase of commissions being abolished in 1871.
So, an officer's son in the first half of the century would probably have gone to a
public school (which means something completely different in the UK to anywhere else in the world), such as Eton, Harrow or Winchester, as his family would have been landed gentry, and well off - toward the end of the period, he would been a lot more likely to have gone to a state school.
But it also depends on the rank of your officer - there was an unwritten, but rigid rule in the Army about marrying:
"A Subaltern may not marry, Captains might marry, Majors should marry, but Colonels must marry."
(there are cases of subalterns being forced to resign from a crack cavalry regiment for marrying)
-so your officer must be of a sufficiently high rank, and have been so for at least the son's age, to to have a child old enough to go to school at all, if that makes sense...
Hope this helps!
(my links, not skim-links, BTW)